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Theorem carden2b 10007
Description: If two sets are equinumerous, then they have equal cardinalities. (This assertion and carden2a 10006 are meant to replace carden 10591 in ZF without AC.) (Contributed by Mario Carneiro, 9-Jan-2013.) (Proof shortened by Mario Carneiro, 27-Apr-2015.)
Assertion
Ref Expression
carden2b (𝐴𝐵 → (card‘𝐴) = (card‘𝐵))

Proof of Theorem carden2b
StepHypRef Expression
1 cardne 10005 . . . . 5 ((card‘𝐵) ∈ (card‘𝐴) → ¬ (card‘𝐵) ≈ 𝐴)
2 ennum 9987 . . . . . . . 8 (𝐴𝐵 → (𝐴 ∈ dom card ↔ 𝐵 ∈ dom card))
32biimpa 476 . . . . . . 7 ((𝐴𝐵𝐴 ∈ dom card) → 𝐵 ∈ dom card)
4 cardid2 9993 . . . . . . 7 (𝐵 ∈ dom card → (card‘𝐵) ≈ 𝐵)
53, 4syl 17 . . . . . 6 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐵) ≈ 𝐵)
6 ensym 9043 . . . . . . 7 (𝐴𝐵𝐵𝐴)
76adantr 480 . . . . . 6 ((𝐴𝐵𝐴 ∈ dom card) → 𝐵𝐴)
8 entr 9046 . . . . . 6 (((card‘𝐵) ≈ 𝐵𝐵𝐴) → (card‘𝐵) ≈ 𝐴)
95, 7, 8syl2anc 584 . . . . 5 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐵) ≈ 𝐴)
101, 9nsyl3 138 . . . 4 ((𝐴𝐵𝐴 ∈ dom card) → ¬ (card‘𝐵) ∈ (card‘𝐴))
11 cardon 9984 . . . . 5 (card‘𝐴) ∈ On
12 cardon 9984 . . . . 5 (card‘𝐵) ∈ On
13 ontri1 6418 . . . . 5 (((card‘𝐴) ∈ On ∧ (card‘𝐵) ∈ On) → ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴)))
1411, 12, 13mp2an 692 . . . 4 ((card‘𝐴) ⊆ (card‘𝐵) ↔ ¬ (card‘𝐵) ∈ (card‘𝐴))
1510, 14sylibr 234 . . 3 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐴) ⊆ (card‘𝐵))
16 cardne 10005 . . . . 5 ((card‘𝐴) ∈ (card‘𝐵) → ¬ (card‘𝐴) ≈ 𝐵)
17 cardid2 9993 . . . . . 6 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
18 id 22 . . . . . 6 (𝐴𝐵𝐴𝐵)
19 entr 9046 . . . . . 6 (((card‘𝐴) ≈ 𝐴𝐴𝐵) → (card‘𝐴) ≈ 𝐵)
2017, 18, 19syl2anr 597 . . . . 5 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐴) ≈ 𝐵)
2116, 20nsyl3 138 . . . 4 ((𝐴𝐵𝐴 ∈ dom card) → ¬ (card‘𝐴) ∈ (card‘𝐵))
22 ontri1 6418 . . . . 5 (((card‘𝐵) ∈ On ∧ (card‘𝐴) ∈ On) → ((card‘𝐵) ⊆ (card‘𝐴) ↔ ¬ (card‘𝐴) ∈ (card‘𝐵)))
2312, 11, 22mp2an 692 . . . 4 ((card‘𝐵) ⊆ (card‘𝐴) ↔ ¬ (card‘𝐴) ∈ (card‘𝐵))
2421, 23sylibr 234 . . 3 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐵) ⊆ (card‘𝐴))
2515, 24eqssd 4001 . 2 ((𝐴𝐵𝐴 ∈ dom card) → (card‘𝐴) = (card‘𝐵))
26 ndmfv 6941 . . . 4 𝐴 ∈ dom card → (card‘𝐴) = ∅)
2726adantl 481 . . 3 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐴) = ∅)
282notbid 318 . . . . 5 (𝐴𝐵 → (¬ 𝐴 ∈ dom card ↔ ¬ 𝐵 ∈ dom card))
2928biimpa 476 . . . 4 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ dom card) → ¬ 𝐵 ∈ dom card)
30 ndmfv 6941 . . . 4 𝐵 ∈ dom card → (card‘𝐵) = ∅)
3129, 30syl 17 . . 3 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐵) = ∅)
3227, 31eqtr4d 2780 . 2 ((𝐴𝐵 ∧ ¬ 𝐴 ∈ dom card) → (card‘𝐴) = (card‘𝐵))
3325, 32pm2.61dan 813 1 (𝐴𝐵 → (card‘𝐴) = (card‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wss 3951  c0 4333   class class class wbr 5143  dom cdm 5685  Oncon0 6384  cfv 6561  cen 8982  cardccrd 9975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-er 8745  df-en 8986  df-card 9979
This theorem is referenced by:  card1  10008  carddom2  10017  cardennn  10023  cardsucinf  10024  pm54.43lem  10040  nnadju  10238  nnadjuALT  10239  ficardun  10241  ackbij1lem5  10263  ackbij1lem8  10266  ackbij1lem9  10267  ackbij2lem2  10279  carden  10591  r1tskina  10822  cardfz  14011
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